An electron moves in a circle of radius$\mathbf{r}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{29}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{11}}\mathbf{\text{m}}$with speed $\mathbf{2}\mathbf{.}\mathbf{19}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\frac{\mathbf{\text{m}}}{\mathbf{\text{s}}}$. Treat the circular path as a current loop with a constant current equal to the ratio of the electron’s charge magnitude to the period of the motion. If the circle lies in a uniform magnetic field of magnitude $\mathbf{B}\mathbf{}\mathbf{=}\mathbf{}\mathbf{7}\mathbf{.}\mathbf{10}$mT, what is the maximum possible magnitude of the torque produced on the loop by the field?

a) Radius of a circle, $r=5.29\times {10}^{-11}\text{m}$

b) Velocity of a charge, $V=2.19\times {10}^6\text{m/s}$ .

c) The magnetic field, $B=7.10\text{mT or}7.10\times {10}^{-3}\text{T}$

The current flowing in a conductor is as follows:

$I=\frac{q}{T}$ …… (i)

Here,$q$is the amount of charge flow,$T$is the time taken for the flow.

The time taken by a circular body to complete a revolution is as follows:

$T=\frac{V}{2\pi r}$ …… (ii)

Here,$V$is the velocity of the body,$2\pi r$is the radius of the circular path.

The torque acting at a point inside a magnetic field is as follows:

$\tau =NiABsin\theta$ ……. (iii)

Here, $N$is the number of turns in the coil, $i$is the current of the wire, $A$is the area of the conductor, $B$is the magnetic field, $\theta$is the angle made by the conductor with the magnetic field.

Using these equations (i) and (ii), the equation of torque can be given as follows:

$$\begin{gathered} \tau =N\times \frac{q}{T}\times A\times B\times sin\theta \\ =N\times \frac{q}{2\pi r}\times \pi r^2\times B\times sin\theta \left(\text{QArea of the circular path,}A=\pi r^2\right) \\ =N\times \frac{q\times V\times r}{2}\times B\times sin\theta \end{gathered}$$

As the torque is perpendicular to the radius and magnetic field$\theta =90°$

So, the value of the maximum possible torque can be given using the given data in the above equation as follows:

$$\begin{gathered} \tau =1\times \frac{\left(1.6\times {10}^{-19}\text{C}\right)\times \left(2.19\times {10}^6\frac{\text{m}}{\text{s}}\right)\times \left(5.29\times {10}^{-11}\text{m}\right)}{2}\times \left(7.10\times {10}^{-3}\text{T}\right)\times \text{sin}\left(90°\right) \\ =6.58\times {10}^{-26}\text{N}·\text{m} \end{gathered}$$

Hence, the value of the maximum possible torque is $6.58\times {10}^{-26}\text{N}·\text{m}$.